The Zero Vector Space
Recall from the Vector Spaces page the definition of a Vector Space:
Definition: A nonempty set $V$ is considered a vector space if the two operations: 1. addition of the objects $\mathbf{u}$ and $\mathbf{v}$ that produces the sum $\mathbf{u} + \mathbf{v}$, and, 2. multiplication of these objects $\mathbf{u}$ with a scalar $a$ that produces the product $a \mathbf{u}$, are both defined and the ten axioms below hold. Furthermore, if $V$ is a vector space then the objects in $V$ are called vectors:
1. $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$ (Commutativity of vector addition). |
The simplest vector space that exists is simply the zero vector space, that is the set $\{ 0 \}$ whose only element is $0$ combined with the operations of standard addition and standard scalar multiplication. We will verify that all ten axioms hold for this vector space, much of which is redundant. Let $a, b, \in \mathbb{F}$.
- 1. $0 + 0 = 0$
- 2. $0 + (0 + 0) = (0 + 0) + 0$.
- 3. The zero vector is $0$, that is $0 + 0 = 0$.
- 4. The additive inverse of $0$ is $-0$, that is $0 + (-0) = 0$.
- 5. $a(b0) = (ab)0 = 0$.
- 6. Any scalar $a \in \mathbb{F}$ works as a multiplicative identity, that is $a0 = 0$.
- 7. $a(0 + 0) = a0 + a0 = 0$.
- 8. $(a + b)0 = a0 + b0 = 0$.
- 9. Since $0 + 0 = 0$, we note that $(0 + 0) \in \{ 0 \}$ and so $\{ 0 \}$ is closed under addition.
- 10. Since $a0 = 0$, we note that $(a0) \in \{ 0 \}$ and so $\{ 0 \}$ is closed under scalar multiplication.
Therefore the set $\{ 0 \}$ containing the operations of standard addition and standard multiplication is a vector space since all ten vector space axioms hold.